A new construction of Bol loops of order \(8k\).
Let \(B\) be a loop with center \(Z(B)\), let \(A\) be an Abelian group, and let \(f\colon A\times A\to Z(B)\) satisfy \(f(0,u)=f(u,0)=1\) for all \(u\in A\). It is straightforward to see that \(L:=B\times A\) with the multiplication \((x,u)(y,v):=(xy\cdot f(u,v),u+v)\) is a loop with normal subloop \(B\). For \(L\) to satisfy any identity such as Bol, Moufang, flexible law, etc., it is clearly necessary that \(B\) satisfies the same identity, as \(B\) is a subloop of \(G\). It is straightforward to add conditions on \(f\) in order to obtain necessary and sufficient conditions. The authors mention this in the acknowledgement. They actually only study two very special cases of the general construction. Let \(m\) be a fixed positive integer and put \(\nu_m(i,j):=(i+j-(i+j)^*)/m\), where \(i^*\) is the least nonnegative residue of the integer \(i\bmod m\). Notice that \(\nu_m(i,j)\in\{0,1\}\) if \(0\leq i,j<m\). For the first construction put \(A:=C_m\) and \(f(i,j)=z^{\nu_m(i,j)}\) for an element \(z\in Z(B)\). If \(B\) is Bol, Moufang, or associative, respectively, then so is \(L\). For the second construction take \(m,n\) even positive integers and choose \(r,s,t,z,w\) in \(Z(B)\) such that \(r^{\gcd(n,m)}=s^2=t^2=1\). Put \(A=C_m\times C_n\) and let \[ f(i,\pi,j,\rho):=r^{j\pi}s^{ij\rho}t^{j\pi\rho}z^{\nu_m(i,j)}w^{\nu_n(\pi,\rho)}. \] Assume \(B\) is a Bol loop, then so is \(L\). The Bol loop \(L\) is Moufang, a group, respectively, iff \(B\) is Moufang, or a group, respectively, and (in both cases) \(s=t=1\). \(L\) is a Bruck loop iff \(B\) is a Bruck loop and \(r^2=rst=1\). The authors also determine the nuclei, centrum, center, and the commutator/associator subloop. Moreover, the construction yields all Bol loops of order \(8\), and many of orders \(16\) and \(24\). However, the questions of isomorphy or isotopy are still open for the latter two orders.
scientific article; zbMATH DE number 2173131
scientific article